Let the complex numbers `z_1,z_2,z_3` be the vertices of an equilateral triangle. Let `z_0` be the circumcentre of the triangle. Then prove that `z_1^2+z_2^2+z_3^2= 3z_0^2`.

If z_(1),z_(2)andz_(3) are the vertices of an equilasteral triangle with z_(0) as its circumcentre , then changing origin to z^(0) ,show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=0, where z_(1),z_(2),z_(3), are new complex numbers of the vertices.

If the complex variables z_1,z_2 and origin form an equilateral triangle, prove that : z_1^2+z_2^2-z_1z_2=0 .

The points A,B,C represent the complex numbers z_1,z_2,z_3 respectively on a complex plane & the angle B & C of the triangle ABC are each equal to 1/2 (pi-alpha) . If (z_2-z_3)^2=lambda(z_3-z_1)(z_1-z_2) sin^2( alpha/2) then determine lambda .

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

If z is any complex number, prove that : |z|^2= |z^2| .

Complex numbers z_(1),z_(2)andz_(3) are the vertices A,B,C respectivelt of an isosceles right angled triangle with right angle at C. show that (z_(1)-z_(2))^(2)=2(z_1-z_(3))(z_(3)-z_(2)).

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?

The complex numbers z_1, z_2 and z_3 satisfying (z_1-z_3)/(z_2- z_3)= (1-i sqrt3)/2 are the vertices of triangle, which is :

If z_(1),z_(2)andz_(3) are the affixes of the vertices of a triangle having its circumcentre at the origin. If zis the affix of its orthocentre, prove that Z_(1)+Z_(2)+Z_(3)-Z=0.

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .